Physics Chapter 04

Chapter 4

Motion in Two Dimensions

Motion in Two Dimensions Using + or – signs is not always

sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe

motion Still interested in displacement,

velocity, and acceleration Will serve as the basis of multiple types

of motion in future chapters

Position and Displacement The position of an

object is described by its position vector, r

The displacement of the object is defined as the change in its position Δr = rf - ri

General Motion Ideas In two- or three-dimensional

kinematics, everything is the same as as in one-dimensional motion except that we must now use full vector notation Positive and negative signs are no

longer sufficient to determine the direction

Average Velocity The average velocity

is the ratio of the displacement to the time interval for the displacement

The direction of the average velocity is the direction of the displacement vector, Δr

t r

v

Average Velocity, cont The average velocity between

points is independent of the path taken This is because it is dependent on the

displacement, also independent of the path

Instantaneous Velocity The instantaneous velocity is the

limit of the average velocity as Δt approaches zero The direction of the instantaneous

velocity is along a line that is tangent to the path of the particle’s direction of motion

0limt

d

t dt

r r

v

Instantaneous Velocity, cont The direction of the instantaneous

velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion

The magnitude of the instantaneous velocity vector is the speed The speed is a scalar quantity

Average Acceleration The average acceleration of a

particle as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.

f i

f it t t

v v v

a

Average Acceleration, cont As a particle

moves, Δv can be found in different ways

The average acceleration is a vector quantity directed along Δv

Instantaneous Acceleration The instantaneous acceleration is

the limit of the average acceleration as Δv/Δt approaches zero

0limt

d

t dt

v v

a

Producing An Acceleration Various changes in a particle’s

motion may produce an acceleration The magnitude of the velocity vector

may change The direction of the velocity vector

may change Even if the magnitude remains constant

Both may change simultaneously

Kinematic Equations for Two-Dimensional Motion When the two-dimensional motion

has a constant acceleration, a series of equations can be developed that describe the motion

These equations will be similar to those of one-dimensional kinematics

Kinematic Equations, 2 Position vector

Velocity

Since acceleration is constant, we can also find an expression for the velocity as a function of time: vf = vi + at

ˆ ˆx y r i j

ˆ ˆx y

dv v

dt r

v i j

Kinematic Equations, 3 The velocity

vector can be represented by its components

vf is generally not along the direction of either vi or at

Kinematic Equations, 4 The position vector can also be

expressed as a function of time: rf = ri + vit + ½ at2

This indicates that the position vector is the sum of three other vectors:

The initial position vector The displacement resulting from vi t The displacement resulting from ½ at2

Kinematic Equations, 5 The vector

representation of the position vector

rf is generally not in the same direction as vi or as ai

rf and vf are generally not in the same direction

Kinematic Equations, Components The equations for final velocity and

final position are vector equations, therefore they may also be written in component form

This shows that two-dimensional motion at constant acceleration is equivalent to two independent motions One motion in the x-direction and the

other in the y-direction

Kinematic Equations, Component Equations vf = vi + at becomes

vxf = vxi + axt and vyf = vyi + ayt

rf = ri + vi t + ½ at2 becomes

xf = xi + vxi t + ½ axt2 and yf = yi + vyi t + ½ ayt2

Projectile Motion An object may move in both the x

and y directions simultaneously The form of two-dimensional

motion we will deal with is called projectile motion

Assumptions of Projectile Motion The free-fall acceleration g is

constant over the range of motion And is directed downward

The effect of air friction is negligible With these assumptions, an object

in projectile motion will follow a parabolic path This path is called the trajectory

Verifying the Parabolic Trajectory Reference frame chosen

y is vertical with upward positive Acceleration components

ay = -g and ax = 0

Initial velocity components vxi = vi cos and vyi = vi sin

Verifying the Parabolic Trajectory, cont Displacements

xf = vxi t = (vi cos t yf = vyi t + ½ay t2 = (vi sin t - ½ gt2

Combining the equations gives:

This is in the form of y = ax – bx2 which is the standard form of a parabola

22 2

tan2 cosi

i i

gy x x

v

Analyzing Projectile Motion Consider the motion as the superposition

of the motions in the x- and y-directions The x-direction has constant velocity

ax = 0

The y-direction is free fall ay = -g

The actual position at any time is given by: rf = ri + vit + ½gt2

Projectile Motion Vectors rf = ri + vi t + ½ g t2

The final position is the vector sum of the initial position, the position resulting from the initial velocity and the position resulting from the acceleration

Projectile Motion Diagram

Projectile Motion – Implications The y-component of the velocity is

zero at the maximum height of the trajectory

The accleration stays the same throughout the trajectory

Range and Maximum Height of a Projectile

When analyzing projectile motion, two characteristics are of special interest

The range, R, is the horizontal distance of the projectile

The maximum height the projectile reaches is h

Height of a Projectile, equation The maximum height of the

projectile can be found in terms of the initial velocity vector:

This equation is valid only for symmetric motion

2 2sin

2i iv

hg

Range of a Projectile, equation The range of a projectile can be

expressed in terms of the initial velocity vector:

This is valid only for symmetric trajectory

2 sin 2i ivR

g

More About the Range of a Projectile

Range of a Projectile, final The maximum range occurs at i =

45o

Complementary angles will produce the same range The maximum height will be different

for the two angles The times of the flight will be different

for the two angles

Projectile Motion – Problem Solving Hints Select a coordinate system Resolve the initial velocity into x and y

components Analyze the horizontal motion using

constant velocity techniques Analyze the vertical motion using

constant acceleration techniques Remember that both directions share

the same time

Non-Symmetric Projectile Motion Follow the general

rules for projectile motion

Break the y-direction into parts

up and down or symmetrical back to

initial height and then the rest of the height

May be non-symmetric in other ways

Uniform Circular Motion Uniform circular motion occurs when

an object moves in a circular path with a constant speed

An acceleration exists since the direction of the motion is changing This change in velocity is related to an

acceleration The velocity vector is always tangent to

the path of the object

Changing Velocity in Uniform Circular Motion The change in the

velocity vector is due to the change in direction

The vector diagram shows v = vf - vi

Centripetal Acceleration The acceleration is always

perpendicular to the path of the motion

The acceleration always points toward the center of the circle of motion

This acceleration is called the centripetal acceleration

Centripetal Acceleration, cont The magnitude of the centripetal

acceleration vector is given by

The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

2

C

va

r

Period The period, T, is the time required

for one complete revolution The speed of the particle would be

the circumference of the circle of motion divided by the period

Therefore, the period is 2 r

Tv

Tangential Acceleration The magnitude of the velocity

could also be changing In this case, there would be a

tangential acceleration

Total Acceleration The tangential

acceleration causes the change in the speed of the particle

The radial acceleration comes from a change in the direction of the velocity vector

Total Acceleration, equations The tangential acceleration:

The radial acceleration:

The total acceleration: Magnitude

t

da

dt

v

2

r C

va a

r

2 2r ta a a

Total Acceleration, In Terms of Unit Vectors

Define the following unit vectors

r lies along the radius vector

is tangent to the circle

The total acceleration is

ˆˆ andr

2ˆ ˆt r

d v

dt r

va a a r

Relative Velocity Two observers moving relative to each other

generally do not agree on the outcome of an experiment

For example, observers A and B below see different paths for the ball

Relative Velocity, generalized Reference frame S

is stationary Reference frame S’

is moving at vo This also means

that S moves at –vo relative to S’

Define time t = 0 as that time when the origins coincide

Relative Velocity, equations The positions as seen from the two

reference frames are related through the velocity r’ = r – vo t

The derivative of the position equation will give the velocity equation v’ = v – vo

These are called the Galilean transformation equations

Acceleration in Different Frames of Reference The derivative of the velocity

equation will give the acceleration equation

The acceleration of the particle measured by an observer in one frame of reference is the same as that measured by any other observer moving at a constant velocity relative to the first frame.